

We refer to this as Loris–Verhoeven with shift,
for reasons that will be clarified later.
Furthermore, when \(A=I\)  in PDHG,
we obtain the Douglas–Rachford splitting (DRS) algorithm
[1]}, [2]}, [3]}.
Conversely, the “completion” technique in [4]} shows that
PDHG coincides with DRS applied to a reformulation of the problem.
Similarly, when \(A=I\)  in the primal Condat–Vũ algorithm (REF ),
we obtain a new algorithm and refer to it as
the reduced primal Condat–Vũ algorithm.
Conversely, the reduced primal Condat–Vũ algorithm
reverts to (REF ) via the “completion” trick.
We can also set \(f=0\)  in the reduced Condat–Vũ algorithm
or \(A=I\)  in (REF ),
and obtain the reduced Loris–Verhoeven algorithm with shift:
\(x^{(k+1)} &= x^{(k)}-\tau (z^{(k)}+\nabla h(x^{(k)})) \\z^{(k+1)} &= \mathrm {prox}_{\sigma g^\ast }\big ( (1-\sigma \tau )z^{(k)}+\sigma (x^{(k+1)}-\tau \nabla h(x^{({\color {red}{k}})})\big ).\) 
